introduction to digital signal and system analysis

Introduction to Digital Signal and System Analysis Download free books at... Download free eBooks at bookboon.com2 Weiji Wang Introduction to Digital Signal and System Analysis... Downl

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Introduction to Digital Signal and System Analysis

Download free books at

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Download free eBooks at bookboon.com

2

Weiji Wang

Introduction to

Digital Signal and System Analysis

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Introduction to Digital Signal and System Analysis

ISBN 978-87-403-0158-8

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Contents

Contents

1 Digital Signals and Sampling 8

1.2 Signal representation and processing 8

1.3 Analogue-to-digital conversion 11

1.4 Sampling theorem 11

1.5 Quantization in an analogue-to-digital converter 13

2 Basic Types of Digital Signals 16

2.1 hree basic signals 16

2.2 Other basic signals 18

2.3 Signal shiting, lipping and scaling 20

2.4 Periodic signals 21

2.5 Examples of signal operations 21

3 Time-domain Analysis 25

3.1 Linear time-invariant (LTI) systems 25

3.2 Diference equations 26

3.3 Block diagram for LTI systems 28

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Contents

3.4 Impulse response 29

3.6 Graphically demonstrated convolution 38

4 Frequency Domain Analysis 42

4.1 Fourier series for periodic digital signals 42

4.2 Fourier transform for non-periodic signals 46

4.3 Properties of Fourier transform 49

4.4 Frequency response 50

4.5 Frequency correspondence when sampling rate is given 56

5 Z Domain Analysis 60

5.1 z-transform and inverse z-transform 60

5.2 Relationship between z-transform and Fourier transform 62

5.3 Z as time shit operator 62

5.4 Transfer function 63

5.5 Z-plane, poles and zeros 67

5.6 Stability of a system 71

5.7 Evaluation of the Fourier transform in the z-plane 75

5.8 Characteristics of 1st and 2nd order systems 77

6 Discrete Fourier Transform 88

6.1 Deinition of discrete Fourier transform 88

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Contents

6.2 Properties of DFT 90

6.3 he fast Fourier transform (FFT) 93

7 Spectral Analysis by DFT 100

7.4 Digital spectral analysis 100

7.5 Spectra of harmonics 100

7.6 Spectral leakage 102

9.8 Performance of windows 104

7.6 Applications of digital spectral analysis 107

Bibliography 111

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Preface

Preface

Since the 1990s, digital signals have been increasingly used not only in various industries and engineering equipments but also in everybody’s daily necessities Mobile phones, TV receivers, music CDs, multimedia computing, etc, are the indispensable items in modern life, in which digital formats are taken as a basic form for carrying and storing information

he major reason for the advancement in the use of digital signals is the big leap forward in the popularization of microelectronics and computing technology in the past three decades Traditional analogue broadcast is being widely upgraded to digital A general shit from analogue to digital systems has taken place and achieved unequivocal beneits

in signal quality, transmission eiciency and storage integrity In addition, data management advantage in digital systems has provided users with a very friendly interface A typical example is the popular pull-down manual, easy to ind, make choices and more choices are made available

As marching into the digital era, many people in diferent sectors are quite keen to understand why this has happened and what might be the next in this area hey hope to obtain basic principles about digital signals and associated digital systems Instead of targeting advanced or expert level, they as beginners oten hope to grasp the subject as eicient and efective as possible without undertaking impossible task under usually limited time and efort available

his book is written for those beginners who want to gain an overview of the topic, understand the basic methods and know how to deal with basic digital signals and digital systems No matter the incentive is from curiosity, interest or urgently acquiring needed knowledge for one’s profession, this book is well suited he output standards are equivalent

to university year two which lays a good foundation for further studies or moving on to specialised topics, such as digital ilters, digital communications, discrete time-frequency representation, and time-scale analysis he required mathematics for the reader is basically at pre-university level, actually only junior high schools maths is mainly involved he content

of materials in this book has been delivered to second year engineering and IT students at university for more than 10 years A feature in this book is that the digital signal or system is mainly treated as originally existing in digital form rather than always regarded as an approximation version of a corresponding analogue system which gives a wrong impression that digital signal is poor in accuracy, although many digital signals come from taking samples out of analogue signals

he digital signal and system stand as their own and no need to use the analogue counter part to explain how they work

To help understanding and gaining good familiarity to the topic, it will be very helpful to do some exercises attached to each chapter, which are selected from many and rather minimal in term of work load

Weiji Wang University of Sussex Brighton, England January 2012

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Digital Signals and Sampling

1 Digital Signals and Sampling

1.1 Introduction

Digital signal processing (DSP) has become a common tool for many disciplines he topic includes the methods of dealing with digital signals and digital systems he techniques are useful for all the branches of natural and social sciences which involve data acquisition, analysis and management, such as engineering, physics, chemistry, meteorology, information systems, inancial and social services Before the digital era, signal processing devices were dominated by analogue type

he major reason for DSP advancement and shit from analogue is the extraordinary growth and popularization of digital microelectronics and computing technology

he reason that digital becomes a trend to replace analogue systems, apart from it is a format that microprocessors can be easily used to carry out functions, high quality data storage, transmission and sophisticated data management are the other advantages In addition, only 0s and 1s are used to represent a digital signal, noise can easily be suppressed or removed

he quality of reproduction is high and independent of the medium used or the number of reproduction Digital images are two dimensional digital signals, which represent another wide application of digital signals Digital machine vision, photographing and videoing are already widely used in various areas

In the ield of signal processing, a signal is deined as a quantity which carries information An analogue signal is a signal represented by a continuous varying quantity A digital signal is a signal represented by a sequence of discrete values of

a quantity he digital signal is the only form for which the modern microprocessor can take and exercise its powerful functions Examples of digital signals which are in common use include digital sound and imaging, digital television, digital communications, audio and video devices

To process a signal is to make numerical manipulation for signal samples he objective of processing a signal can be to detect the trend, to extract a wanted signal from a mixture of various signal components including unwanted noise, to look at the patterns present in a signal for understanding underlying physical processes in the real world To analyse a digital system is to ind out the relationship between input and output, or to design a processor with pre-deined functions, such as iltering and amplifying under applied certain frequency range requirements A digital signal or a digital system can be analysed in time domain, frequency domain or complex domain, etc

1.2 Signal representation and processing

Representation of digital signals can be speciic or generic A digital signal is refereed to a series of numerical numbers, such as:

…, 2, 4, 6, 8, …

where 2, 4, 6 are samples and the whole set of samples is called a signal In a generic form, a digital signal can be represented

as time-equally spaced data

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], 2 [ ], 1 [ ], 0 [ ], 1 [ , x − x x x

where -1, 0, 1, 2 etc are the sample numbers, x[0], x[1], x[2], etc are samples he square brackets represent the digital form he signal can be represented as a compact form

∞

<

∞

n

x ] [

In the signal, x[-1], x[1], x[100], etc, are the samples, n is the sample number he values of a digital signal are only being deined at the sample number variable n, which indicates the occurrence order of samples and may be given a speciic unit of time, such as second, hour, year or even century, in speciic applications

We can have many digital signal examples:

- Midday temperature at Brighton city, measured on successive days,

- Daily share price,

- Monthly cost in telephone bills,

- Student number enrolled on a course,

- Numbers of vehicles passing a bridge, etc

Examples of digital signal processing can be given in the following:

Example 1.1 To obtain a past 7 day’s average temperature sequence he averaged temperature sequence for past 7 days is

7

1

]

[ n = x n + x n − + x n − + x n −

For example, if n=0 represents today, the past 7 days average is

7

1

]

0

[ = x + x − + x − + x −

y

where x [ 0 ], x [ − 1 ], x [ − 2 ], represent the temperatures of today, yesterday, the day before yesterday, …;y [ 0 ]

represents the average of past 7 days temperature from today and including today On the other hand,

7

1

]

1

[ = x + x + x − + + x −

y

represents the average of past 7 days temperature observed from tomorrow and including tomorrow, and so on In a shorter form, the new sequence of averaged temperature can be written as

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∑

=

+

−

= 7

1

] 1 [

7

1 ]

[

k

k n x n

y

where x[n] is the temperature sequence signal and y[n] is the new averaged temperature sequence he purpose of average can be used to indicate the trend he averaging acts as a low-pass ilter, in which fast luctuations have been removed as

a result herefore, the sequence y[n] will be smoother than x[n]

Example 1.2 To obtain the past M day simple moving averages of share prices, let x[n] denotes the close price, yM[ n ]

the averaged close price over past M days

( [ ] [ 1 ] [ 2 ] [ 1 ] )

1 ] [ = x n + x n − + x n − + x n − M +

M n

yM

or

∑

=

+

−

k

M x n k

M n y

1

] 1 [

1 ] [

(1.2)

For example, M=20 day simple moving average is used to indicate 20 day trend of a share price M=5, 120, 250 (trading days) are usually used for indicating 1 week, half year and one year trends, respectively Figure 1.1 shows a share’s prices with moving averages of diferent trading days

Figure 1.1 S share prices with moving averages

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1.3 Analogue-to-digital conversion

Although some signals are originally digital, such as population data, number of vehicles and share prices, many practical signals start of in analogue form hey are continuous signals, such as human’s blood pressure, temperature and heart pulses A continuous signal can be irst converted to a proportional voltage waveform by a suitable transducer, i.e the analogue signal is generated hen, for adapting digital processor, the signal has to be converted into digital form by taking samples hose samples are usually equally spaced in time for easy processing and interpretation Figure 1.2 shows

a analogue signal and its digital signal by sampling with equal time intervals he upper is the analogue signal x(t) and the lower is the digital signal sampled at time t = nT, where n is the sample number and T is the sampling interval herefore,

) ( ] [ n x nT

0 20 40 60 80 10 0 -50

0 50

t

0 20 40 60 80 10 0 -50

0 50

t

Figure 1.2 An analogue signal x(t) and digital signal x[n] he upper is the analogue signal and the lower is the digital signal sampled at t = nT

1.4 Sampling theorem

For ease of storage or digital processing, an analogue signal must be sampled into a digital signal he continuous signal

is being taken sample at equal time interval and represented by a set of members First of all, a major question about it

is how oten should an analogue signal be sampled, or how frequent the sampling can be enough to represent the details

of the original signal It is obvious that too oten will cause redundancy which will reduce the processing eiciency and cause an unnecessarily large size of data storage, but too sparse will cause a loss of signal details

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